The Eckhaus instability: from initial to final stages

TL;DR

This paper investigates the nonlinear evolution of the Eckhaus instability for the one-dimensional Ginzburg-Landau equation under small broadband perturbations. The authors perform direct numerical integration on a large periodic domain, initializing with a stationary unstable periodic solution plus a noise-like spectrum $\psi_0(x)=\sqrt{1-k_0^2}\,e^{ik_0x}+A\sum_{n=-N}^{N}e^{i(n\Delta k x+\varphi_{0n})}$ and monitoring $\psi$, its Fourier spectrum $\psi_k$, and the Lyapunov functional $\mathcal{F}$, while decomposing $\psi$ into real and imaginary parts to handle phase slips. They identify four nonlinear stages: rapid decay of stable perturbations, latent spectral sharpening toward unstable modes, a phase-slip interval with multiple $2\pi$ phase shifts and sharp decreases in $\mathcal{F}$, and a final slow relaxation to a single selected wavenumber, here $k=\Delta k$, though the final state generally depends on the initial perturbations. The results deliver a unified nonlinear picture of wavenumber selection in pattern formation, elucidating how phase slips and spectrum evolution guide systems from unstable periodic states to stable, Eckhaus-selected patterns through a quantifiable Lyapunov-tracking framework.

Abstract

A systematic analysis of the Eckhaus instability in the one-dimensional Ginzburg-Landau equation is presented. The analysis is based on numerical integration of the equation in a large (xt)-domain. The initial conditions correspond to a stationary, unstable spatially periodic solution perturbed by "noise." The latter consists of a set of spatially periodic modes with small amplitudes and random phases. The evolution of the solution is examined by analyzing and comparing the dynamics of three key characteristics: the solution itself, its spatial spectrum, and the value of the Lyapunov functional. All calculations exhibit four distinct, mutually agreed, well-defined regimes: (i) rapid decay of stable perturbations; (ii) latent changes, when the solution and the Lyapunov functional undergo minimal alterations while the Fourier spectrum concentrates around the most unstable perturbations; (iii) a phase-slip period, characterized by a sharp decrease in the Lyapunov functional; (iv) slow relaxation to a final stable state.

Figures (10)

• Figure 1: Profiles of Re$\,\psi(x,t)$, Im$\,\psi(x,t)$, and $|\psi(x,t)|$.
• Figure 2: (a) Temporal evolution at $0\leq t \leq 70$ of the Lyapunov functional for the GL equation with the initial conditions given by Eq. \ref{['eq:psi0_num']}. (b) Zoom of the initial part of this profile.
• Figure 3: (a) The initial profile of the modulus of $\psi$ at $t=0$ (marked by number 1) is shown, along with its phase normalized by $\pi$ (marked by number 2). The phase, denoted as $\varphi$, is confined to the range $-\pi \leq \varphi \leq \pi$. If the continuous phase value exceeds this domain, it is manually adjusted by adding (or subtracting) the appropriate integer multiples of $2\pi$. This adjustment introduces discontinuities in the corresponding profile. (b) A log plot displays the moduli of the coefficients from the Fourier series expansion of $\psi(x,0)$.
• Figure 4: The same profiles as those in Fig. \ref{['fig:t0']} at $t=2$.
• Figure 5: The same profiles as those in Fig. \ref{['fig:t0']} at $t=8$.